Theses, Dissertations, and Articles of my Students

Paul Sinclair

Metrics on Bundle Spaces and Harmonic Gauss Maps
Doctoral Dissertation, December 1991

The aim of this work is to define a pseudoriemannian metric on the Grassmannian bundles Gn over a pseudoriemannian manifold M, with n < dim M+1, and to explore the relationship between the harmonicity of an immersion in and the harmonicity of its Gauss map into Gn M. In the process, a general method is developed for defining, à la Sasaki, a metric on the total space of a fiber bundle, given metrics on the base space and on the standard fiber, and a connection on the bundle. This metric exists only when the structure group of the bundle is reducible to a subgroup of the isotropy group of the metric on the standard fiber. In this case, the metric obtained also depends on how this reduction of the structure group is achieved.

Once the metric is defined, two pseudoriemannian connections on the total space are considered. One is the Levi-Civita connection. The other is also pseudoriemannian, and is closely related to the connections on the base space and the standard fiber. The relationship between the harmonicity of a map into the total space and of its projection pf into the base space is studied for both of these connections. The result obtained is that, for the Levi-Civita connection, the harmonicity of implies that of pf only when the connection in the bundle is flat. When applied to Grassmannian bundles and Gauss maps, this has strong implications for certain string theories in physics. [PDF]

Kamielle Freeman

A Historical Overview of Connections in Geometry
Master's Thesis, May 2011

This thesis is an attempt to untangle/clarify the modern theory of connections in Geometry. Towards this end a historical approach was taken and original as well as secondary sources were used. An overview of the most important historical developments is given as well as a modern look at how the various definitions of connection are related.

I hope to clear up some of the confusions surrounding a connection in Differential Geometry; or, at least some of the things that confused me when I was trying to figure out just what exactly a connection is. In particular, 1) How is a covariant derivative operator related to a connection? 2)How is parallel transport related to the previous two notions? 3) What was the first definition of a connection? 4) What is a connection in the most general sense? I hope that I answer all of these questions satisfactorily in the following pages. I have also provided a chart of the heirarchy of connections via the use of Lie subgroups. [PDF]

Justin M. Ryan

Geometry of Horizontal Bundles and Connections
Doctoral Dissertation, May 2014

An Ehresmann connection on a fiber bundle E over M is defined by prescribing a suitable horizontal subbundle H of the tangent bundle TE over E. For a horizontal bundle to be suitable, it must have the property of horizontal path lifting. This ensures that the horizontal bundle determines a system of parallel transport between any two fibers of E.

The main result of this dissertation is a geometric characterization of the horizontal bundles on E that have horizontal path lifting, and hence are connections. In particular, it is shown that a horizontal bundle has horizontal path lifting if and only if its horizontal spaces are bounded away from the vertical spaces, uniformly along fibers of E.

In order for a horizontal bundle to admit a system of parallel transport or have holonomy, it must be a connection. However, certain other geometric properties that are usually attributed to connections are actually properties of arbitrary horizontal bundles. These properties are studied in the case when E is either a vector bundle or tangent bundle, accordingly. [PDF]

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